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Theorem 2moswapv 2691
 Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap 2706 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexexvw 2690. (Revised by Wolf Lammen, 2-Oct-2023.)
Assertion
Ref Expression
2moswapv (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2moswapv
StepHypRef Expression
1 nfe1 2151 . . . 4 𝑦𝑦𝜑
21nfmov 2619 . . . 4 𝑦∃*𝑥𝑦𝜑
3 nfe1 2151 . . . . 5 𝑥𝑥(∃𝑦𝜑𝜑)
43nfmov 2619 . . . 4 𝑥∃*𝑦𝑥(∃𝑦𝜑𝜑)
51, 2, 4moexexlem 2688 . . 3 ((∃*𝑥𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦𝑥(∃𝑦𝜑𝜑))
65expcom 417 . 2 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥(∃𝑦𝜑𝜑)))
7 19.8a 2178 . . . . 5 (𝜑 → ∃𝑦𝜑)
87pm4.71ri 564 . . . 4 (𝜑 ↔ (∃𝑦𝜑𝜑))
98exbii 1849 . . 3 (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑𝜑))
109mobii 2606 . 2 (∃*𝑦𝑥𝜑 ↔ ∃*𝑦𝑥(∃𝑦𝜑𝜑))
116, 10syl6ibr 255 1 (∀𝑥∃*𝑦𝜑 → (∃*𝑥𝑦𝜑 → ∃*𝑦𝑥𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ∃wex 1781  ∃*wmo 2596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598 This theorem is referenced by:  2euswapv  2692  2rmoswap  3700
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